# An analysis of the boundary layer in the 1D surface Cauchy–Born model

Kavinda Jayawardana; Christelle Mordacq; Christoph Ortner; Harold S. Park

- Volume: 47, Issue: 1, page 109-123
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topJayawardana, Kavinda, et al. "An analysis of the boundary layer in the 1D surface Cauchy–Born model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 109-123. <http://eudml.org/doc/273244>.

@article{Jayawardana2013,

abstract = {The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.},

author = {Jayawardana, Kavinda, Mordacq, Christelle, Ortner, Christoph, Park, Harold S.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {surface-dominated materials; surface Cauchy–Born rule; coarse-graining; surface Cauchy-Born rule},

language = {eng},

number = {1},

pages = {109-123},

publisher = {EDP-Sciences},

title = {An analysis of the boundary layer in the 1D surface Cauchy–Born model},

url = {http://eudml.org/doc/273244},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Jayawardana, Kavinda

AU - Mordacq, Christelle

AU - Ortner, Christoph

AU - Park, Harold S.

TI - An analysis of the boundary layer in the 1D surface Cauchy–Born model

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 1

SP - 109

EP - 123

AB - The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

LA - eng

KW - surface-dominated materials; surface Cauchy–Born rule; coarse-graining; surface Cauchy-Born rule

UR - http://eudml.org/doc/273244

ER -

## References

top- [1] X. Blanc, C. Le Bris, and P.-L. Lions, From molecular models to continuum mechanics. Arch. Ration. Mech. Anal.164 (2002) 341–381. Zbl1028.74005MR1933632
- [2] X. Blanc, C. Le Bris, and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM : M2AN 39 (2005) 797–826. Zbl1330.74066MR2165680
- [3] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems. Math. Models Methods Appl. Sci.17 (2007) 985–1037. Zbl1205.82036MR2337429
- [4] A. Braides, A.J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal.180 (2006) 151–182. Zbl1093.74013MR2210908
- [5] R.C. Cammarata, Surface and interface stress effects in thin films. Prog. Surf. Sci.46 (1994) 1–38.
- [6] S. Cuenot, C. Frétigny, S. Demoustier-Champagne and B. Nysten, Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69 (2004) 165410.
- [7] J. Diao, K. Gall and M.L. Dunn, Surface-stress-induced phase transformation in metal nanowires. Nat. Mater.2 (2003) 656–660.
- [8] M. Dobson and M. Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error. ESAIM : M2AN 43 (2009) 591–604. Zbl1165.81414MR2536250
- [9] M. Dobson, M. Luskin and C. Ortner, Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids58 (2010) 1741–1757. Zbl1200.74005MR2742030
- [10] W. E and P. Ming, Cauchy–Born rule and the stability of crystalline solids : static problems. Arch. Ration. Mech. Anal. 183 (2007) 241–297. Zbl1106.74019MR2278407
- [11] M. Farsad, F.J. Vernerey and H.S. Park, An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials. Int. J. Numer. Methods Eng.84 (2010) 1466–1489. Zbl1202.74168MR2777999
- [12] G. Friesecke and F. Theil, Validity and failure of the Cauchy–Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci.12 (2002) 445–478. Zbl1084.74501MR1923388
- [13] W. Gao, S.W. Yu and G.Y. Huang, Finite element characterization of the size-dependent mechanical behaviour in nanosystems. Nanotechnol.17 (2006) 1118–1122.
- [14] M.E. Gurtin and A. Murdoch, A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal.57 (1975) 291–323. Zbl0326.73001MR371223
- [15] J. He and C.M. Lilley, The finite element absolute nodal coordinate formulation incorporated with surface stress effect to model elastic bending nanowires in large deformation. Comput. Mech.44 (2009) 395–403. Zbl1166.74038
- [16] A. Javili and P. Steinmann, A finite element framework for continua with boundary energies. part I : the two-dimensional case. Comput. Methods Appl. Mech. Eng. 198 (2009) 2198–2208. Zbl1227.74075MR2519130
- [17] H. Liang, M. Upmanyu and H. Huang, Size-dependent elasticity of nanowires : nonlinear effects. Phys. Rev. B 71 (2005) R241–403.
- [18] W. Liang, M. Zhou and F. Ke, Shape memory effect in Cu nanowires. Nano Lett.5 (2005) 2039–2043.
- [19] C. Ortner and F. Theil, Nonlinear elasticity from atomistic mechanics. E-prints arXiv:1202.3858v3 (2012).
- [20] H.S. Park, Surface stress effects on the resonant properties of silicon nanowires. J. Appl. Phys. 103 (2008) 123504.
- [21] H.S. Park, Quantifying the size-dependent effect of the residual surface stress on the resonant frequencies of silicon nanowires if finite deformation kinematics are considered. Nanotechnol. 20 (2009) 115701.
- [22] H.S. Park and P.A. Klein, Surface Cauchy–Born analysis of surface stress effects on metallic nanowires. Phys. Rev. B 75 (2007) 085408.
- [23] H.S. Park and P.A. Klein, A surface Cauchy–Born model for silicon nanostructures. Comput. Methods Appl. Mech. Eng.197 (2008) 3249–3260. Zbl1159.74312MR2440608
- [24] H.S. Park and P.A. Klein, Surface stress effects on the resonant properties of metal nanowires : the importance of finite deformation kinematics and the impact of the residual surface stress. J. Mech. Phys. Solids56 (2008) 3144–3166. Zbl1175.74009
- [25] H.S. Park, K. Gall and J.A. Zimmerman, Shape memory and pseudoelasticity in metal nanowires. Phys. Rev. Lett. 95 (2005) 255504.
- [26] H.S. Park, P.A. Klein and G.J. Wagner, A surface Cauchy–Born model for nanoscale materials. Int. J. Numer. Methods Eng.68 (2006) 1072–1095. Zbl1128.74005MR2274690
- [27] H.S. Park, W. Cai, H.D. Espinosa and H. Huang, Mechanics of crystalline nanowires. MRS Bull.34 (2009) 178–183.
- [28] H.S. Park, M. Devel and Z. Wang, A new multiscale formulation for the electromechanical behavior of nanomaterials. Comput. Methods Appl. Mech. Eng.200 (2011) 2447–2457. Zbl1230.74082MR2803638
- [29] H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Convection-diffusion-reaction and flow problems, 2nd edition. Springer Series in Comput. Math. 24 (2008). Zbl1155.65087MR2454024
- [30] P. Rosakis, Continuum surface energy from a lattice model. E-prints arXiv:1201.0712 (2012). Zbl1305.74074MR3268168
- [31] L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems. Math. Mod. Methods Appl. Sci.21 (2011) 777–817. Zbl1268.49016MR2795506
- [32] B. Schmidt. On the passage from atomic to continuum theory for thin films. Arch. Ration. Mech. Anal.190 (2008) 1–55. Zbl1156.74028MR2434899
- [33] J.-H. Seo, Y. Yoo, N.-Y. Park, S.-W. Yoon, H. Lee, S. Han, S.-W. Lee, T.-Y. Seong, S.-C. Lee, K.-B. Lee, P.-R. Cha, H.S. Park, B. Kim and J.-P. Ahn, Superplastic deformation of defect-free au nanowires by coherent twin propagation. Nano Lett.11 (2011) 3499–3502.
- [34] A.V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions. Multiscale Model. Simul.9 (2011) 905–932. Zbl1246.74065MR2831585
- [35] C.Q. Sun, B.K. Tay, X.T. Zeng, S. Li, T.P. Chen, J. Zhou, H.L. Bai and E.Y. Jiang, Bond-order-bond-length-bond-strength (bond-OLS) correlation mechanism for the shape-and-size dependence of a nanosolid. J. Phys. : Condens. Matter 14 (2002) 7781–7795.
- [36] F. Theil, A proof of crystallization in two dimensions. Commun. Math. Phys.262 (2006) 209–236. Zbl1113.82016MR2200888
- [37] F. Theil, Surface energies in a two-dimensional mass-spring model for crystals. ESAIM : M2AN 45 (2011) 873–899. Zbl1269.82065MR2817548
- [38] G. Yun and H.S. Park, A multiscale, finite deformation formulation for surface stress effects on the coupled thermomechanical behavior of nanomaterials. Comput. Methods Appl. Mech. Eng.197 (2008) 3337–3350. Zbl1159.74315MR2440611
- [39] G. Yun and H.S. Park, Surface stress effects on the bending properties of fcc metal nanowires. Phys. Rev. B 79 (2009) 195421.
- [40] J. Yvonnet, H. Le Quang and Q.-C. He, An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput. Mech.42 (2008) 119–131. Zbl1188.74076MR2404304

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.